Skeleton of Berkovich spaces Pedro A. Castillejo 1.12.2016 Abstract This is the sixth talk of the Research Seminar of the Arithmetic Geometry group (FU Berlin), which is on the Winter term of 2016-17. In the seminar we want to understand the paper [NX16]. In this talk I'll try to explain the skeleton of a Berkovich space, and I will try to use this information to draw some 1-dimensional examples. Sadly I'm not able yet to make drawings in the computer, so the only skeleton that will appear is this one: \Tate m'a ´ecritde son c^ot´esur ses histoires de courbes el- liptiques, et pour me demander si j'avais des id´eessur une d´efinitionglobale des vari´et´esanalytiques sur des corps valu´es complets. Je dois avouer que je n'ai pas du tout compris pourquoi ses r´esultats sugg´ereraient l'existence d'une telle d´efinition, et suis encore sceptique. Je n'ai pas non plus l'impression d'avoir rien compris `ason th´eor`eme,qui ne fait qu'exhiber par des formules brutales un certain isomorphisme de groupes analytiques; on con¸coitque d'autres formules tout aussi explicites en donneraeient un autre pas plus mauvais (sauf preuve du contraire!). " A. Grothendieck to J.-P. Serre, 18.08.19591 1\Tate has written to me about his elliptic curves stuff, and has asked me if I had any ideas for a global definition of analytic varieties over complete valuation fields. I must admit that I have absolutely not understood why his results suggest the existence of such a definition, and I remain skeptical. Nor do I have the impression of having understood his theorem at all; it does nothing more than exhibit, via brute formulas, a certain isomorphism of analytic groups; one could conceive that other equally explicit formulas might give another one which would be no worse than his (until proof to the contrary!)." 1 Birational, monomial and divisorial points of Berkovich spaces Setting: here R is a complete discrete valuation ring, with residue field k and fraction field K. We also set X to be a connected regular separated K-scheme of finite type. Recall that we can define its Berkovich analytification, denoted Xan, as the set x 2 X; and j · jK(x) a valuation on the residue field K(x) x; j · jK(x) of the point x extending the valuation j · jK together with a topology and a sheaf of rings. Today we will only be interested in Xan as a topological space, whose properties were described in the talk of Wouter. This Berkovich analytification comes naturally with a map ι : Xan ! X; which by the definition of the Berkovich topology is continuous. Remark 1. If x 2 X is a closed point, the fibre ι−1(x) consists in just one point, namely (x; j · jK(x)), because K(x) is an algebraic extension (since x is closed) of a complete discrete valued field, and therefore there is a unique valuation j · jK(x) extending j · jK . In the last week, we defined the set of birational points of Xan, denoted Xbir, as the fibre ι−1(η), where η is the generic point of X, endowed with the subspace topology. Remark 2. If X is a curve (for example the affine line over K), then Xan n Xbir is in bijection with the set of closed points of X. For the affine line, those are precisely the type I points. Before proceeding, recall that a sncd-model of X is a regular flat separated scheme X of finite type over R, together with an isomorphism between the generic fibre XK and X, such that the special fibre Xk is a divisor with strict normal crossings. As usual (SGA 5, 3.1.5, pg. 24), we say that a divisor D on X has strict normal crossings if P given global sections (fi) such that D = i div(fi), we have that for every x 2 Supp(D), the local restrictions (fi)x lying in mx form a part of a regular system of parameters for the local ring OX;x. For example, for curves, this means that X is regular and that its singular points are ordinary double points. We also defined the monomial points, that will be very important in this talk, so we also recall the definition. A monomial point is constructed from the data (X ; (Ej1 ;:::;Ejr ); (α1; : : : ; αr); ξ); where •X is an sncd-model of X, Pm • the Ei's are some of the prime divisors of the special fibre Xk = i=1 NiEi, the Ni's being the multiplicities of the irreducible components in the special fibre, r • the r-tuple α 2 R≥0 are just some real numbers such that Nj1 α1 + ··· + Njr αr = 1, and • ξ is the generic point of one of the connected components of Ej1 \···\ Ejr . 2 We explained in the previous talk that every regular function f 2 OX ,ξ can be written in O[X ,ξ as a series X β cβT ; β2Nr with cβ a unit or 0 in O[X ,ξ, not necessarily in a unique way (c.f. [MN13, Lemma 2.4.4]). And from this data, we were able to associate a unique valuation defined as v := v : f 7! minfα · βjβ 2 r; c 6= 0g α (X ;(Ej1 ;:::;Ejr );(α1,...,αr),ξ) N β which is independent of the choice of the representation of f as a power series and that restricted to j · jK gives our original valuation (c.f. [MN13, Prop. 2.4.6]). This valuation yields a point an (η; exp(−vα)) in X . The set of all points of this form (i.e. running through all the choices of models, divisors, parameters, etc.) is called the set of monomial points, and denoted Xmon. Given a monomial point x, we say that it is divisorial if there exists a model X such that x is associated to (X ;Ei; 1=Ni; ξ), being ξ the generic point of Ei. Proposition 3. The monomial point associated to the data (X ; (Ej1 ;:::;Ejr ); α; ξ) is divisorial if and only if all the αi are in Q≥0. The proof is in [MN13, Prop. 2.4.11]. Here we show the phenomena with a concrete example, so that we can clarify this a little bit more. Example 4. Let X = Spec K[T1;T2]=(π − T1T2) and X = Spec R[T1;T2]=(π − T1T2), where π is a uniformizer of R. Then the special fibre Xk = Spec k[T1;T2]=(T1T2) consists of two lines, that we call E1 and E2, intersecting in a point that we call O. Now we can consider the monomial point x 2 Xmon associated to the data (X ; (E1;E2); (1=2; 1=2);O); which is not a divisorial point with respect to this model X . But, according to the previous proposition, this is a divisorial point, so let's try to find a model X 0 so that x is divisorial (i.e. associated to a divisor) with respect to that model. For this, we blow up X in O and we get X 0 !X Ef1 ! E1 Ef2 ! E2 E ! O where E is the exceptional divisor mapping to the node O, and Efi is the strict transform of Ei. Then, one sees that the divisorial point associated to (X 0;E) is precisely x. Remark 5. The following table explains the relation between this classification of points, and the classification of points for curves: Xdiv Xmon Xbir Xan Type II Types II and III Types II, III and IV Types I,. , IV In higher dimension, there are a lot of different points. Por example, in dimension 2, there are points in Xan n Xbir whose image under the analytification map ι is not closed (think for example in ξ, the generic point of an irreducible divisor, together with any valuation). 3 The dual intersection complex P In the above setting, given a sncd-model X of X with special fibre Xk = i2I NiEi, we want to define a simplicial complex that encodes important information of X . We will actually be interested just in the underlying topological space of this complex, so we construct it directly. T For any non-empty subset J ⊂ I, we denote EJ := j2J Ej. Definition 6. The dual intersection complex of the model X , denoted j∆(Xk)j, is defined as follows: • The faces: for all d 2 N, we stablish a bijection fSimplices of dimension dg ! fConnected components of EJ ; where jJj = d + 1g τ 7−! Cτ : 0 • The glueing: given τ; τ simplices of j∆(Xk)j, we stablish 0 τ ⊂ τ () Cτ 0 ⊂ Cτ : Remark 7. Note that vertices of j∆(Xk)j correspond o irreducible components of Xk. Remark 8. The points of j∆(Xk)j can be seen as couples (ξ; β), where ξ corresponds to the generic point of an intersection of r distinct irreducible components of Xk, and β is an element of ( r ) o r X ∆ξ := x 2 R>0j xi = 1 : i=1 This couple is called the barycentric coordinates, and we will see later how to relate them with the data that defines monomial points. Example 9. If we have a Tate curve of type In (this is the Kodaira-N´eronclassification), with n ≥ 2, then it has a minimal model X where the special fibre Xk is given by a loop of n copies of P1.